Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32878 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#32881 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C24×C3×D4 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C24×C3×C4 ◁ G | |
5942 conjugacy classes of subgroups; only the 438 normal subgroups are shown.
This group has 5942 conjugacy classes of subgroups — too many to draw. Only its 438 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(4,5)(6,7)(8,9)(10,11)(12,13,15,18)(14,16,19,17);(13,16)(14,19)(17,18);(5,7);(9,11)174 conjugacy classes — view the full character table.
| Fitting subgroup of | D3×#32881, #72253290 |
| maximal subgroup of | D3×#32881, #72253290, C3×#6628842 |
| maximal normal subgroup of | D3×#32881, #72253290, C3×#6628842 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 25869);